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    Re: Impossible lunar example
    From: Frank Reed
    Date: 2010 Aug 30, 20:45 -0700

    John, you wrote:
    "Frank, I’m having trouble following this thread without going back over many posts. You recently wrote..."

    Yeah, I don't blame you. The section that quoted as 'recently wrote' was actually from late 2004 which I had quoted back just so folks would know that this has been discussed before --and quite some time ago at that!

    You wrote:
    "I get lost in this last sentence. Neither shifting the moon’s altitude down to 40d nor clearing the lunar distance changes the azimuth difference (what I call the relative bearing angle, RBA). What am I missing. Or do I completely misunderstand the topic here??"

    It's a small mistake but a critical one. Suppose you're trying to assess the sensitivity of the problem of clearing a lunar to the value of the Moon's altitude. The inputs to the problem are the altitudes of the two bodies and the observed lunar distance (ignoring for now the possibility of variations in refraction and the value of the Moon's HP). Let's take the simple case of the Moon's and Sun's altitudes (altitudes of the centers) both at 45 degrees and the measured lunar distance ALSO at 45 degrees (center-to-center distance). To assess the sensitivity, we want to consider changing the Moon's altitude only, while holding the lunar distance and the Sun's altitude constant. If I change the Moon's altitude, then the "relative bearing angle" will necessarily change. Try it: what is the relative bearing for the case as given? Now try it again with the Moon's altitude shifted to 44d. And at 46d? You can work this out using any solution for clearing lunars that appeals to you. But seriously, you need to try it to see it.

    So why is this an important distinction? In years past on NavList, some people have thought that they could understand the sensitivity of the clearing problem to changes (errors) in the objects' altitudes, just by looking at the changes in the altitude corrections. They reasoned that the lunar distance problem was not terribly sensitive to errors in altitude because the altitude corrections (the common altitude correction tables in the Nautical Almanac) change rather slowly with altitude. If you enter the altitude correction table for stars with 45d 00' or with 45d 03', there is no difference in the altitude correction. And this is an important factor. But the catch is that changing the altitude ALSO changes the geometry of the triangle. And that change in shape alters the fraction of the altitude correction that lies along the actual lunar arc. These TWO factors, one the sensitivity of the altitude corrections themselves to altitude changes, and the other the sensitivity of the shape of the triangle to altitude changes, work together to yield the net sensitivity of the lunar clearing process to changes/errors in the observed altitudes. By the way, what I am describing here can also be viewed as a straight-forward calculus problem: take the equation for clearing lunars in some appropriate form (as I demonstrated in Mystic in June) and take partial derivatives with respect to the altitudes. What you'll find are the error-sensitivity equations that I first described for NavList way back in the fall of 2004: errors as large as 6'*tan(dist)/cos(h_moon) in the Moon's altitude or as large as 6'*sin(dist)/cos(h_body) in the other body's altitude will yield insignificant errors in the final results --no larger than 0.1' in the clearing process (these equations are somewhat approximate and would need to be modified for altitudes lower than about 10 degrees). We end up with two nice compact equations containing a wealth of information about the nature of the process of clearing lunars. And I would point out again that these equations were apparently unknown historically though there are definite hints that some navigators and nautical astronomers understood the general qualitative behavior though they did not have equations for the quantitative behavior.

    For the issue at hand, inconsistent triangles, the interesting point is that the math STILL WORKS even if the triangle seems to be broken. I STRONGLY encourage anyone interested in this issue to work the first of the "impossible" cases in the Tables Requisite from 1781. With the Moon 88d high, the Sun 5d high and the measured LD at 89.9d, the observations are apparently inconsistent. It's an "impossible triangle". But as I have said repeatedly in this topic, it doesn't matter. The clearing problem is quite insensitive to large errors in the Moon's altitude, especially when the distance is close to 90d OR when the Moon is very high (in this example we have BOTH). To convince yourself, shift the Moon's altitude to 85d and clear it again. You should get almost exactly the same result for the cleared lunar distance.

    Note that none of this applies to the second "impossible" case in the Tables Requisite. It's clearly worse. In that one, there's no valid navigational data that can come out of the problem, but as a teaching example, there's no issue (EXCEPT for those few students who try to experiment with it further, and for them it would have been quite frustrating).

    -FER


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